Download A concise introduction to pure mathematics by Martin Liebeck PDF

By Martin Liebeck

Accessible to all scholars with a valid historical past in highschool arithmetic, A Concise advent to natural arithmetic, Fourth Edition offers the most primary and gorgeous rules in natural arithmetic. It covers not just general fabric but in addition many fascinating themes now not often encountered at this point, corresponding to the speculation of fixing cubic equations; Euler’s formulation for the numbers of corners, edges, and faces of an outstanding item and the 5 Platonic solids; using best numbers to encode and decode mystery info; the idea of the way to match the sizes of 2 limitless units; and the rigorous conception of limits and non-stop functions.

New to the Fourth Edition

  • Two new chapters that function an creation to summary algebra through the idea of teams, masking summary reasoning in addition to many examples and applications
  • New fabric on inequalities, counting tools, the inclusion-exclusion precept, and Euler’s phi functionality
  • Numerous new workouts, with options to the odd-numbered ones

Through cautious motives and examples, this well known textbook illustrates the facility and sweetness of simple mathematical thoughts in quantity idea, discrete arithmetic, research, and summary algebra. Written in a rigorous but available variety, it maintains to supply a powerful bridge among highschool and higher-level arithmetic, permitting scholars to check extra complicated classes in summary algebra and analysis.

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A concise introduction to pure mathematics

Available to all scholars with a legitimate heritage in highschool arithmetic, A Concise creation to natural arithmetic, Fourth variation offers probably the most primary and gorgeous principles in natural arithmetic. It covers not just general fabric but in addition many fascinating themes no longer often encountered at this point, reminiscent of the speculation of fixing cubic equations; Euler’s formulation for the numbers of corners, edges, and faces of a great item and the 5 Platonic solids; using best numbers to encode and decode mystery info; the idea of the way to check the sizes of 2 endless units; and the rigorous concept of limits and non-stop features.

Extra info for A concise introduction to pure mathematics

Example text

2, and state it here. 1 Let n be a positive integer. If x is a positive real number, then there is exactly one positive real number y such that yn = x. If x, y are as in the statement, we adopt the familiar notation 1 y = xn . 1 1 Thus, for example, 5 2 is the positive square root of 5, and 5 7 is the unique positive real number y such that y7 = 5. We can extend this notation to define rational powers of positive reals as follows. Let x > 0. Integer powers xm (m ∈ Z) are defined in the familiar way: if m > 0 then xm = xx .

34 A CONCISE INTRODUCTION TO PURE MATHEMATICS Notice that |x| just measures the distance from the point x on the real line to the origin 0. Thus, for example, the set of values of x such that |x| ≤ 2 consists of all x between −2 and 2, which we summarize as −2 ≤ x ≤ 2. 10 Let a, b ∈ R with b > 0. For which values of x is the inequality |x − a| ≤ b satisfied? Answer When x ≥ a, the inequality says x − a ≤ b; that is, x ≤ a + b. And when x < a, the inequality says a − x ≤ b; that is, x ≥ a − b. So the range of values of x satisfying the inequality is a − b ≤ x ≤ a + b.

2. 14 is not as innocent as it looks. For example, √ it tells you that among all rectangles with a given area A, the square of side A has the smallest perimeter (see Exercise 8 at the end of the chapter). It is also the case n = 2 of a famous and very important inequality known as the “Arithmetic-Geometric Mean Inequality,” which states that if n is a positive integer and a1 , . . , an are positive real numbers, then 1 (a1 a2 · · · an )1/n ≤ (a1 + a2 + · · · + an ). 1) (The right hand side is the “arithmetic mean” of the numbers a1 , .

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